CHAPTER 5 the Black–Scholes model
In this Chapter the foundations of derivatives theory: delta hedging and no arbitrage the derivation of the Black–Scholes partial differential equation the assumptions that go into the Black–Scholes equation how to modify the equation for commodity and currency options
2
(5.2)
ELIMINATION OF RISK: DELTA HEDGING:
If we choose
∂V ∆= ∂S
(5.3)
then the randomness is in randomness is generally termed hedging, whether that randomness is due to fluctuations in the stock market or the outcome of a horse race. The perfect elimination of risk, by exploiting correlation between two instruments (in this case an option and its underlying) is generally called delta hedging.
NO ARBITRAGE: After choosing the quantity delta as suggested above, we hold a portfolio whose value changes by the amount:
∂V 1 2 2 ∂ 2V dΠ = dt + σ S dt 2 ∂t 2 ∂S
∂V 1 2 2 ∂ V ∂V + σ S + rS − rV = 0 2 ∂t 2 ∂S ∂S
2
(5.6)
d Π = dV − ∆ dS
From Itˆo we have:
∂V ∂V 1 2 2 ∂ V dV = dt + dS + σ S dt 2 ∂t ∂S 2 ∂S
2
Thus the portfolio changes by:
∂V ∂V 1 2 2 ∂ V dΠ = dt + dS + σ S dt − ∆ dS 2 ∂t ∂S 2 ∂S
Π = V (S , t ) − ∆S
(5.1)
We will assume that the underlying follows a lognormal random walk
dS = µ Sdt + σ SdX
It is natural to ask how the value of the portfolio changes from time t to t + dt. The change in the portfolio value is due partly to the change in the option value and partly to the change in the underlying:
A VERY SPECIAL PORTFOLIO: Use Π to denote the value of a portfolio of one long option position and a short position in some quantity , ∆ delta, of the underlying:
(5.4)
This change is completely riskless:
d Π = r Π dt
This is an example of the no arbitrage principle.
(5.5)
THE BLACK–SCHOLES EQUATION： ： Substituting (5.1), (5.3) and (5.4) into (5.5) we get that